The Casimir Energy For Scalar Field With Mixed Boundary Condition

TL;DR

This paper calculates the first-order radiative correction to the Casimir energy for scalar fields with mixed boundary conditions, employing a systematic renormalization method and the Box Subtraction Scheme, and compares results across different boundary conditions.

Contribution

It introduces a position-dependent renormalization approach and applies the Box Subtraction Scheme to compute Casimir energy with mixed boundary conditions in  theory.

Findings

Casimir energy results are consistent with physical expectations.

The sign and magnitude of Casimir energy vary with boundary conditions.

Comparison shows differences between Dirichlet, Neumann, mixed, and periodic cases.

Abstract

In the present study, the first-order radiative correction to the Casimir energy for massive and massless scalar fields confined with mixed boundary conditions (Dirichlet-Neumann) between two points in \phi^4 theory was computed. Two issues in performing the calculations in this work are essential: to renormalize the bare parameters of the problem, a systematic method was employed, allowing all influences from the boundary conditions to be imported in all elements of the renormalization program. This idea yields our counterterms appeared in the renormalization program to be position-dependent. Using the Box Subtraction Scheme (BSS) as a regularization technique is the other noteworthy point in the calculation. In this scheme, by subtracting the vacuum energies of two similar configurations from each other, regularizing divergent expressions and their removal process were significantly facilitated. All the obtained answers for the Casimir energy with the mixed boundary condition were consistent with well-known physical grounds. We also compared the Casimir energy for massive scalar field confined with four types of boundary conditions (Dirichlet, Neumann, mixed of them and Periodic) in 1+1 dimensions with each other, and the sign and magnitude of their values were discussed.